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Ted123
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HallsofIvy said:Are you sure the function is defined as you say? Yes, as given, f is NOT continuous because it is not defined at x= -1 and at x= 1.
If the function were defined as
[tex]f(x)= \left{\begin{array}{cc} -1, & x<-1 \\ x, & -1\le x\le 1 \\ 1, & x> 1\end{array}[/tex]
then it would be continuous and the first and second derivatives exactly as you say.
Derivatives are mathematical tools used to measure the rate of change of a function at a specific point. They are used in various branches of mathematics, especially in calculus, to solve problems related to optimization, rates of change, and slope of graphs.
To sketch a graph using derivatives, you need to find the critical points of the function, which are the points where the derivative is equal to zero or does not exist. Then, you can use the first and second derivative tests to determine the nature of these points and the shape of the graph.
A local maximum is a point on a graph where the function has the greatest value in a specific interval, while a local minimum is a point where the function has the smallest value in that interval. They are also known as relative extrema.
To find the derivative of a function, you can use various methods such as the power rule, product rule, quotient rule, and chain rule. These methods involve taking the limit of the difference quotient as the change in the independent variable approaches zero.
The relationship between derivatives and integrals is described by the fundamental theorem of calculus, which states that the derivative of a function is the inverse operation of the integral of the same function. In other words, the integral of the derivative of a function is equal to the original function.